As described in Section , the linear map algorithm is
related
to the stochastic EnKF. The only difference between these two algorithms lies
in the way they exploit structure for the purpose of regularization.
While stochastic EnKFs exploit decay of correlation between the variables with increasing spatial distance, regularization in the stochastic map filter enforces sparsity in the map \(S^{\mathcal {X}}\) , based on conditional independence, decay of correlation, and local likelihood structure (see Sections  and ). In the stochastic map filter, the sparsity structure is parameterized by the distance \(r\)  that defines the set of input state variables for each component in (REF ), and by the number of components \(j \le n\)  where \(S^{\mathcal {X}}\)  departs from the identity map (see Remark REF ). In the stochastic EnKF, covariance localization is applied by multiplying the forecast covariance elementwise with a Gaspari–Cohn tapering function that has an optimally tuned localization radius [1]}.
While the linear maps and the stochastic EnKF behave similarly with larger ensemble sizes—where less regularization is needed—the results may differ for small \(M\) .
